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In-Depth Information
Ψ
dqs
is the rotating flux matrix
Ψ
dqs
=
T
(
θ
)
Ψ
s
=
L
dqs
i
dqs
+
Ψ
dqm
(8)
Ω
dqs
is the speed matrix.
Ω
dqs
=
d
T
(
θ
)
dt
T
−
1
(
θ
)
(9)
In (8), the inductance matrix is
⎡
⎣
⎤
⎦
L
d
1
0
L
13
00
0
L
q
1
0
L
13
0
L
13
0
L
d
3
00
0
L
13
0
L
q
3
0
0000
L
ls
L
s
T
−
1
(
θ
)=
L
dqs
=
T
(
θ
)
(10)
The flux matrix of magnet is
Ψ
m
=[0
ψ
m
1
0
ψ
m
3
0]
T
Ψ
dqm
=
T
(
θ
)
(11)
The speed matrix is
⎡
⎤
01000
−
10000
00030
00
⎣
⎦
Ω
dqs
=
d
T
(
θ
)
dt
T
−
1
(
θ
)=
ω
(12)
300
00000
−
2.4 Electromagnetic Torque Equation
With the virtual displacement method in the electric motor theory, the electro-
magnetic torque is
T
e
=
P
1
∂
L
s
∂θ
i
s
+
∂
Ψ
m
∂θ
s
s
2
i
i
(13)
From (7)-(12), the electromagnetic torque
T
e
is calculated,
ψ
m
1
i
q
1
+3
ψ
m
3
i
q
3
+(
L
d
1
−
L
q
1
)
i
d
1
i
q
1
T
e
=
5
P
2
(14)
+3(
L
d
3
−
L
q
3
)+2
L
13
(
i
d
1
i
q
3
−
i
q
1
i
d
3
)
When the field oriented control,
i
d
= 0 is adopted, as
T
e
=
5
P
2
(
ψ
m
1
+3
ψ
m
3
i
q
3
)
(15)
The torque is composed of traditional foundational torque 5
P/
2
ψ
m
1
i
q
1
and three
harmonic torque 15
P/
2
ψ
m
3
i
q
3
. So the three harmonic can be applied to increase
the output torque.
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